﻿certain 
  Problems 
  relating 
  to 
  the 
  Potential. 
  199 
  

  

  culled 
  .r 
  , 
  t 
  r_ 
  1? 
  * 
  l3 
  ,v. 
  2 
  , 
  a?_a, 
  we 
  find 
  

  

  A 
  = 
  ,v 
  , 
  and 
  

  

  

  A!= 
  3 
  (arj 
  «-i)- 
  l2 
  fa 
  #- 
  2 
  ), 
  A 
  8 
  = 
  * 
  12 
  2 
  

  

  6 
  ' 
  

  

  ! 
  2(«] 
  + 
  .''-1 
  — 
  2« 
  ) 
  #s 
  + 
  -''-2 
  — 
  2*o 
  

   Aj 
  ~ 
  3 
  24 
  

  

  

  . 
  _ 
  * 
  a 
  -1- 
  .'-_ 
  2 
  — 
  2.v 
  a?j 
  + 
  a?_i 
  — 
  2cT 
  

  

  

  24 
  6 
  

  

  The 
  B's 
  are 
  deduced 
  from 
  the 
  A's 
  by 
  merely 
  writing 
  y 
  for 
  x 
  

   throughout. 
  Thus 
  from 
  (14) 
  when 
  f 
  = 
  0, 
  rj 
  = 
  1, 
  

  

  5 
  1 
  

  

  .<■ 
  = 
  x 
  Q 
  — 
  g 
  Ox 
  + 
  .r_i 
  — 
  2*o) 
  + 
  J9 
  ( 
  tl 
  '2 
  + 
  *-a 
  - 
  2ii 
  'o) 
  

  

  5 
  1 
  

  

  — 
  g 
  (j/i-y-0 
  + 
  g-fr* 
  — 
  #-2). 
  - 
  . 
  • 
  (16) 
  

  

  Similarly 
  

  

  5 
  1 
  

  

  y 
  = 
  y* 
  — 
  5 
  (yi 
  + 
  y-i 
  - 
  2 
  /a) 
  + 
  12 
  (^ 
  + 
  y- 
  a 
  - 
  2#>) 
  

   5 
  1 
  

  

  + 
  g 
  (*i 
  - 
  *_i) 
  - 
  ^fe 
  - 
  .r_ 
  2 
  ). 
  . 
  . 
  . 
  (17) 
  

  

  By 
  these 
  formula) 
  a 
  point 
  is 
  found 
  upon 
  a 
  new 
  stream-line 
  

   (77=1) 
  corresponding 
  to 
  a 
  given 
  value 
  of 
  f. 
  And 
  there 
  

   would 
  be 
  no 
  difficulty 
  in 
  carrying 
  the 
  approximation 
  further 
  

   if 
  desired. 
  

  

  As 
  an 
  example 
  of 
  the 
  kind 
  of 
  problem 
  to 
  which 
  these 
  

   results 
  might 
  be 
  applied, 
  suppose 
  that 
  by 
  observation 
  or 
  other- 
  

   wise 
  we 
  know 
  the 
  form 
  of 
  the 
  upper 
  stream-line 
  constituting 
  

   part 
  of 
  the 
  free 
  surface 
  when 
  liquid 
  falls 
  steadily 
  over 
  a 
  

   two-dimensional 
  weir. 
  Since 
  the 
  velocity 
  is 
  known 
  at 
  every 
  

   point 
  of 
  the 
  free 
  surface, 
  we 
  are 
  in 
  a 
  position 
  to 
  determine 
  

   % 
  along 
  this 
  stream-line, 
  and 
  thus 
  to 
  apply 
  the 
  formula? 
  so 
  as 
  

   to 
  find 
  interior 
  stream-lines 
  in 
  succession. 
  

  

  2 
  lin 
  (with 
  interchange 
  of 
  f 
  and 
  rj) 
  we 
  could 
  find 
  what 
  

   forms 
  are 
  admissible 
  for 
  the 
  second 
  coating 
  of 
  a 
  two- 
  

   dimensional 
  condenser, 
  in 
  order 
  that 
  the 
  charge 
  upon 
  the 
  

   first 
  coating, 
  given 
  in 
  size 
  and 
  shape, 
  may 
  have 
  a 
  given 
  value 
  

   at 
  every 
  point. 
  

  

  